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Level 541

Sequences & Series Formula


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Test for an Arithmetic Progression
T₂ - T₁ = T₃ - T₂ = d
Teat for a Geometric Progression
T₂ ÷ T₁ = T₃ ÷ T₂ = r
The term of an Arithmetic progression
Tn = a + (n-1) d
Tn = arⁿ⁻¹
The term of an Geometric progression
Sn = ½n( a+ l)
The sum of an Arithmetic progression if you know the last term
Sn = ½n( 2a+(n-1) d)
The sum of an Arithmetic progression if you DON'T know the last term
Sn = a(1 - rⁿ)/1 - r
The sum of a Geometric progression if r<1
Sn = a(rⁿ - 1)/ r- 1
The sum of a Geometric progression if r>1
Sum to Infinity
Only a geometric progression
Compound Interest
A = P(1+r/n)^(nt)
u1
first term of a sequence
n
number of terms in a sequence
l
last term of a sequence
d
common difference
R
common ratio
un
value of the nth term
Sn
sum of the first n terms
Sinfinity
sum to infinity
Arithmetic sequence
a sequence of numbers that has a common difference between every two consecutive terms
arithmetic sequence: recursive formula
find term from previous term
arithmetic sequence: general formula
find any term using n
geometric sequence
a sequence of numbers that has a common ratio between every two consecutive terms
- find general formula
when asked to find when a certain value reached in a sequence
Series
sum of the terms of a sequence
Arithmetic series
Sum of the terms of a sequence
geometric series
sum of the terms of an arithmetic sequence
divergent series
a series that approaches infinity
convergent series
a series that is approaching a limit
find limit
u1 / (1-r)
limit only limit if
can be approached from both sides
= L
notation of limit
graph function
find limit with GDC
Sequence
A set of numbers arranged in a special order or pattern.
pn(x)=1-((x^2)/2!)+((x^4)/4!)-((x^6)/6!)+....((x^n)/n!)
Maclauren series of cos(x)
divergent
harmonic series 1/N
MacLauren series of [1/(1-x)]
1 + x + x^2 + x^3 + x^4 ...+ x^n
1+x+(x^2)/(2!)+.....(x^n)/(n!)
MacLauren series of e^x
x-((x^3)/3!)+((x^5)/5!)-((x^7)/7!)+...
Maclaurin series of sin x
Alternating Harmonic Series
The series [ ( (-1) ^ n+1 ) / n ] ... converges from 1 to infinity
MacLauren series of ln(1 + x)
x - (x²/2) + (x³/3) - (x⁴/4) + .... + ((-1ⁿ) * (xⁿ/ n))
Nth Term Test
If Limit as K approaches infinity, then the Series of K Diverges.
P-Series
Sum where 1/(n^p) converges if and only if p > 1
Geometric Series Convergence Test
If IrI<1, then the geometric series Σar^n converges.
Integral Test
take integral and evaluate, if it goes to a number it is convergent, if there is ∞in it, its divergent
Alternating Series Test
If An is positive, the series ∑(-1)An converges if & only if..
Comparison Test
after comparing remember to take the limit as n→∞ of an/bn and for convergence it must lie on the interval 0<L<∞
1-(x^2)+(x^4)- ... +(-1)ⁿ((xⁿ)/n)
Maclaurin Series of ln(1+x^2)
Ratio Test
limit as n→∞ of an+1/an
limit comparison test
Let A and B be series with positive terms such that p = limit A/B (to infinity) and 0 < p < infinity, then both series converge or both diverge
Lagrange Error Bound
Error = M/((n+1)!) * (x-a)^(n+1) where M is the highest value on the interval
nth -degree Taylor polynomial
Pn(x) = ∑ (fk(a) / k! ) (x-a)^k
Maclaurin Series of 1/x-1
1 + x + x^2 + x^3....(xⁿ)
Alternating Series Error Bound
If Σa sub n is an alternating series in which |a sub n+1| < |a sub n| for all k, then the error in the nth partial sum is no larger than |a sub n+1|.
Absolute Convergence
∑An is convergent as well as ∑|An|
interval of convergence
set of x values for which power series converges
Taylor Series
If a function is differentiable infinitely many times in some interval around x=a, then the Taylor Series centered at a for f is:
Power Series
do the ratio test, then find the interval of convergence and where the origin is
1-x+x^2-x^3...(-1^n)(x^n)
McClaurin series for 1/(x+1)
x-(x^3/3)+(x^5)/5-....((-1)^x)*(x^(n+1))/(2n+1)
McClaurin series for acrtan(x)
Conditional Convergence
∑An is convergent but ∑|An| diverges
a=r^k
Geometric Sequence
Reciprocal of Factors
∑(1/n!) converges to e
arithmetic recursive
an = an-1 + d
arithmetic explicit
an = a₁ + (n-1)d
Sn = ½n(a₁+an)
arithmetic sum of the series
arithmetic sum of series
Sn = ½n(2a₁ + (n-1)d)
geometric recursive
gn = (gn-1)r
Geometric Explicit
a(n)=a x r to the n - 1
Sn = (g₁(1-r^n))/1 - r
sum of a finite geometric series
Sn = (g₁(r^n - 1))/r - 1
sum of a finite geometric series when r>1
Sn = g₁/(1-r)
sum of an infinite geometric series